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Fixed Points and Stability of the Cauchy Functional Equation in
-Algebras
Fixed Point Theory and Applications volume 2009, Article number: 809232 (2009)
Abstract
Using the fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms in -algebras and Lie
-algebras and of derivations on
-algebras and Lie
-algebras for the Cauchy functional equation.
1. Introduction and Preliminaries
The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' Theorem was generalized by Aoki [3] for additive mappings and by Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias [4] has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability of functional equations. A generalization of the Th. M. Rassias theorem was obtained by Găvruţa [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias' approach. The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [6–19]).
-
J.
M. Rassias [20, 21] following the spirit of the innovative approach of Th. M. Rassias [4] for the unbounded Cauchy difference proved a similar stability theorem in which he replaced the factor
by
for
with
(see also [22] for a number of other new results).
We recall a fundamental result in fixed point theory.
Let be a set. A function
is called a generalized metric on
if
satisfies
(1) if and only if
;
(2) for all
;
(3) for all
.
Let be a complete generalized metric space and let
be a strictly contractive mapping with Lipschitz constant
. Then for each given element
, either

for all nonnegative integers or there exists a positive integer
such that
(1);
(2)the sequence converges to a fixed point
of
;
(3) is the unique fixed point of
in the set
;
(4) for all
.
This paper is organized as follows. In Sections 2 and 3, using the fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms in -algebras and of derivations on
-algebras for the Cauchy functional equation.
In Sections 4 and 5, using the fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms in Lie -algebras and of derivations on Lie
-algebras for the Cauchy functional equation.
2. Stability of Homomorphisms in
-Algebras
Throughout this section, assume that is a
-algebra with norm
and that
is a
-algebra with norm
.
For a given mapping , we define

for all and all
.
Note that a -linear mapping
is called a homomorphism in
-algebras if
satisfies
and
for all
.
We prove the generalized Hyers-Ulam stability of homomorphisms in -algebras for the functional equation
.
Theorem 2.1.
Let be a mapping for which there exists a function
such that



for all and all
. If there exists an
such that
for all
, then there exists a unique
-algebra homomorphism
such that

for all .
Proof.
Consider the set

and introduce the generalized metric on :

It is easy to show that is complete.
Now we consider the linear mapping such that

for all .
By [23, Theorem 3.1],

for all .
Letting and
in (2.2), we get

for all . So

for all . Hence
.
By Theorem 1.1, there exists a mapping such that
(1) is a fixed point of
, that is,

for all . The mapping
is a unique fixed point of
in the set

This implies that is a unique mapping satisfying (2.12) such that there exists
satisfying

for all .
(2) as
. This implies the equality

for all .
(3), which implies the inequality

This implies that the inequality (2.5) holds.
It follows from (2.2) and (2.15) that

for all . So

for all .
Letting in (2.2), we get

for all and all
. By a similar method to above, we get

for all and all
. Thus one can show that the mapping
is
-linear.
It follows from (2.3) that

for all . So

for all .
It follows from (2.4) that

for all . So

for all .
Thus is a
-algebra homomorphism satisfying (2.5), as desired.
Corollary 2.2.
Let and
be nonnegative real numbers, and let
be a mapping such that



for all and all
. Then there exists a unique
-algebra homomorphism
such that

for all .
Proof.
The proof follows from Theorem 2.1 by taking

for all . Then
and we get the desired result.
Theorem 2.3.
Let be a mapping for which there exists a function
satisfying (2.2), (2.3), and (2.4). If there exists an
such that
for all
, then there exists a unique
-algebra homomorphism
such that

for all .
Proof.
We consider the linear mapping such that

for all .
It follows from (2.10) that

for all . Hence,
.
By Theorem 1.1, there exists a mapping such that
(1) is a fixed point of
, that is,

for all . The mapping
is a unique fixed point of
in the set

This implies that is a unique mapping satisfying (2.33) such that there exists
satisfying

for all .
(2) as
. This implies the equality

for all .
(3), which implies the inequality

which implies that the inequality (2.30) holds.
The rest of the proof is similar to the proof of Theorem 2.1.
Corollary 2.4.
Let and
be nonnegative real numbers, and let
be a mapping satisfying (2.25), (2.26), and (2.27). Then there exists a unique
-algebra homomorphism
such that

for all .
Proof.
The proof follows from Theorem 2.3 by taking

for all . Then
and we get the desired result.
3. Stability of Derivations on
-Algebras
Throughout this section, assume that is a
-algebra with norm
.
Note that a -linear mapping
is called a derivation on
if
satisfies
for all
.
We prove the generalized Hyers-Ulam stability of derivations on -algebras for the functional equation
.
Theorem 3.1.
Let be a mapping for which there exists a function
such that


for all and all
. If there exists an
such that
for all
. Then there exists a unique derivation
such that

for all .
Proof.
By the same reasoning as the proof of Theorem 2.1, there exists a unique involutive -linear mapping
satisfying (3.3). The mapping
is given by

for all .
It follows from (3.2) that

for all . So

for all . Thus
is a derivation satisfying (3.3).
Corollary 3.2.
Let and
be nonnegative real numbers, and let
be a mapping such that


for all and all
. Then there exists a unique derivation
such that

for all .
Proof.
The proof follows from Theorem 3.1 by taking

for all . Then
and we get the desired result.
Theorem 3.3.
Let be a mapping for which there exists a function
satisfying (3.1) and (3.2). If there exists an
such that
for all
, then there exists a unique derivation
such that

for all .
Proof.
The proof is similar to the proofs of Theorems 2.3 and 3.1.
Corollary 3.4.
Let and
be nonnegative real numbers, and let
be a mapping satisfying (3.7) and (3.8). Then there exists a unique derivation
such that

for all .
Proof.
The proof follows from Theorem 3.3 by taking

for all . Then
and we get the desired result.
4. Stability of Homomorphisms in Lie
-Algebras
A -algebra
, endowed with the Lie product
on
, is called a Lie
-algebra (see [9–11]).
Definition 4.1.
Let and
be Lie
-algebras. A
-linear mapping
is called aLie
-algebra homomorphism if
for all
.
Throughout this section, assume that is a Lie
-algebra with norm
and that
is a
-algebra with norm
.
We prove the generalized Hyers-Ulam stability of homomorphisms in Lie -algebras for the functional equation
.
Theorem 4.2.
Let be a mapping for which there exists a function
satisfying (2.2) such that

for all . If there exists an
such that
for all
, then there exists a unique Lie
-algebra homomorphism
satisfying (2.5).
Proof.
By the same reasoning as the proof of Theorem 2.1, there exists a unique -linear mapping
satisfying (2.5). The mapping
is given by

for all .
It follows from (4.1) that

for all . So

for all .
Thus is a Lie
-algebra homomorphism satisfying (2.5), as desired.
Corollary 4.3.
Let and
be nonnegative real numbers, and let
be a mapping satisfying (2.25) such that

for all . Then there exists a unique Lie
-algebra homomorphism
satisfying (2.28).
Proof.
The proof follows from Theorem 4.2 by taking

for all . Then
and we get the desired result.
Theorem 4.4.
Let be a mapping for which there exists a function
satisfying (2.2) and (4.1). If there exists an
such that
for all
, then there exists a unique Lie
-algebra homomorphism
satisfying (2.30).
Proof.
The proof is similar to the proofs of Theorems 2.3 and 4.2.
Corollary 4.5.
Let and
be nonnegative real numbers, and let
be a mapping satisfying (2.25) and (4.5). Then there exists a unique Lie
-algebra homomorphism
satisfying (2.38).
Proof.
The proof follows from Theorem 4.4 by taking

for all . Then
and we get the desired result.
Definition 4.6.
A -algebra
, endowed with the Jordan product
for all
, is called aJordan
-algebra (see [25]).
Definition 4.7.
Let and
be Jordan
-algebras.
(i)A -linear mapping
is called a Jordan
-algebra homomorphism if
for all
.
(ii)A -linear mapping
is called a Jordan derivation if
for all
.
Remark 4.8.
If the Lie products in the statements of the theorems in this section are replaced by the Jordan products
, then one obtains Jordan
-algebra homomorphisms instead of Lie
-algebra homomorphisms.
5. Stability of Lie Derivations on
-Algebras
Definition 5.1.
Let be a Lie
-algebra. A
-linear mapping
is called aLie derivation if
for all
.
Throughout this section, assume that is a Lie
-algebra with norm
.
We prove the generalized Hyers-Ulam stability of derivations on Lie -algebras for the functional equation
.
Theorem 5.2.
Let be a mapping for which there exists a function
satisfying (3.1) such that

for all . If there exists an
such that
for all
. Then there exists a unique Lie derivation
satisfying (3.3).
Proof.
By the same reasoning as the proof of Theorem 2.1, there exists a unique involutive -linear mapping
satisfying (3.3). The mapping
is given by

for all .
It follows from (5.1) that

for all . So

for all . Thus
is a derivation satisfying (3.3).
Corollary 5.3.
Let and
be nonnegative real numbers, and let
be a mapping satisfying (3.7) such that

for all . Then there exists a unique Lie derivation
satisfying (3.9).
Proof.
The proof follows from Theorem 5.2 by taking

for all . Then
and we get the desired result.
Theorem 5.4.
Let be a mapping for which there exists a function
satisfying (3.1) and (5.1). If there exists an
such that
for all
, then there exists a unique Lie derivation
satisfying (3.11).
Proof.
The proof is similar to the proofs of Theorems 2.3 and 5.2.
Corollary 5.5.
Let and
be nonnegative real numbers, and let
be a mapping satisfying (3.7) and (5.5). Then there exists a unique Lie derivation
satisfying (3.12).
Proof.
The proof follows from Theorem 5.4 by taking

for all . Then
and we get the desired result.
Remark 5.6.
If the Lie products in the statements of the theorems in this section are replaced by the Jordan products
, then one obtains Jordan derivations instead of Lie derivations.
References
Ulam SM: A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8. Interscience, New York, NY, USA; 1960:xiii+150.
Hyers DH: On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of the United States of America 1941,27(4):222–224. 10.1073/pnas.27.4.222
Aoki T: On the stability of the linear transformation in Banach spaces. Journal of the Mathematical Society of Japan 1950, 2: 64–66. 10.2969/jmsj/00210064
Rassias ThM: On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society 1978,72(2):297–300. 10.1090/S0002-9939-1978-0507327-1
Găvruţa P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. Journal of Mathematical Analysis and Applications 1994,184(3):431–436. 10.1006/jmaa.1994.1211
Park C-G: On the stability of the linear mapping in Banach modules. Journal of Mathematical Analysis and Applications 2002,275(2):711–720. 10.1016/S0022-247X(02)00386-4
Park C-G: Modified Trif's functional equations in Banach modules over a
-algebra and approximate algebra homomorphisms Journal of Mathematical Analysis and Applications 2003,278(1):93–108. 10.1016/S0022-247X(02)00573-5
Park C-G: On an approximate automorphism on a
-algebra Proceedings of the American Mathematical Society 2004,132(6):1739–1745. 10.1090/S0002-9939-03-07252-6
Park C-G: Lie
-homomorphisms between Lie
-algebras and Lie
-derivations on Lie
-algebras Journal of Mathematical Analysis and Applications 2004,293(2):419–434. 10.1016/j.jmaa.2003.10.051
Park C-G: Homomorphisms between Lie
-algebras and Cauchy-Rassias stability of Lie
-algebra derivations Journal of Lie Theory 2005,15(2):393–414.
Park C-G: Homomorphisms between Poisson
-algebras Bulletin of the Brazilian Mathematical Society 2005,36(1):79–97. 10.1007/s00574-005-0029-z
Park C-G: Hyers-Ulam-Rassias stability of a generalized Euler-Lagrange type additive mapping and isomorphisms between
-algebras Bulletin of the Belgian Mathematical Society. Simon Stevin 2006,13(4):619–632.
Park C: Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras. Fixed Point Theory and Applications 2007, 2007:-15.
Park C, Cho YS, Han M-H: Functional inequalities associated with Jordan-von Neumann-type additive functional equations. Journal of Inequalities and Applications 2007, 2007:-13.
Park C, Cui J: Generalized stability of -ternary quadratic mappings. Abstract and Applied Analysis 2007, 2007:-6.
Park C-G, Hou J: Homomorphisms between -algebras associated with the Trif functional equation and linear derivations on -algebras. Journal of the Korean Mathematical Society 2004,41(3):461–477.
Rassias ThM: Problem 16; 2, Report of the 27th International Symposium on Functional Equations. Aequationes Mathematicae 1990,39(2–3):292–293, 309.
Rassias ThM: The problem of S. M. Ulam for approximately multiplicative mappings. Journal of Mathematical Analysis and Applications 2000,246(2):352–378. 10.1006/jmaa.2000.6788
Rassias ThM: On the stability of functional equations in Banach spaces. Journal of Mathematical Analysis and Applications 2000,251(1):264–284. 10.1006/jmaa.2000.7046
Rassias JM: On approximation of approximately linear mappings by linear mappings. Journal of Functional Analysis 1982,46(1):126–130. 10.1016/0022-1236(82)90048-9
Rassias JM: On approximation of approximately linear mappings by linear mappings. Bulletin des Sciences Mathématiques 1984,108(4):445–446.
Rassias JM: Solution of a problem of Ulam. Journal of Approximation Theory 1989,57(3):268–273. 10.1016/0021-9045(89)90041-5
Cădariu L, Radu V: Fixed points and the stability of Jensen's functional equation. Journal of Inequalities in Pure and Applied Mathematics 2003,4(1, article 4):1–7.
Diaz JB, Margolis B: A fixed point theorem of the alternative, for contractions on a generalized complete metric space. Bulletin of the American Mathematical Society 1968,74(2):305–309. 10.1090/S0002-9904-1968-11933-0
Fleming RJ, Jamison JE: Isometries on Banach Spaces: Function Spaces, Chapman & Hall/CRC Monographs and Surveys in Pure & Applied Mathematics. Volume 129. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2003:x+197.
Acknowledgment
This work was supported by Korea Research Foundation Grant KRF-2008-313-C00041.
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Park, C. Fixed Points and Stability of the Cauchy Functional Equation in -Algebras.
Fixed Point Theory Appl 2009, 809232 (2009). https://doi.org/10.1155/2009/809232
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DOI: https://doi.org/10.1155/2009/809232
Keywords
- Linear Mapping
- Functional Equation
- Unique Mapping
- Stability Problem
- Contractive Mapping